 Hello and welcome to this screencast on section 9.4, the cross product. This screencast is going to cover length and direction. Just as we found for the dot product in last section, the cross product of two vectors provides us with useful geometric information. Consider two vectors, U and V, drawn with their tails touching, where the angle between them is denoted by theta, as we have here. These two vectors form the following parallelogram. And using properties of the cross product, we can show that the length of the cross product of U and V is equal to the product of the length of U, the length of V, and sine of theta. See your textbook for details about how to arrive at this. Note that the area of the parallelogram here is equal, also equal to this expression, the length of U times the length of V, sine theta. We can visualize this by chopping this left part of the parallelogram, this triangle, and transposing it right here. This gives us a rectangle that has the same area as the original parallelogram. The area of this rectangle is given by its width, which is this length here. This length here is the length of the vector V times sine theta, multiplied with the rectangle's length. And since we took this piece of the vector U, which is now over here, that means the length of the rectangle from here to here is the length of the vector U. But length times width gives us this expression. So this shows that the length of a cross product of two vectors U and V is equal to the area of the parallelogram determined by U and V. As a vector, we now understand the length of the cross product of U and V. Now let's turn our attention to the direction of this cross product. Recall that the dot product of two vectors tells us information about the direction of a vector. In particular, we want to consider two vectors here, U and V, that are not parallel. If we dot product the vector U with the vector that results from the cross product of U cross V, we see that this gives us zero. Properties of the dot product here tell us that these two vectors must be orthogonal then. And similarly, we can show that the vector V and the vector U cross V are orthogonal. And this tells us that the cross product of two vectors U and V gives us a new vector that is orthogonal to both U and V. There is one more geometric result regarding the cross product. Seven three vectors, U, V and W and R three that are not coplanar, meaning they do not all belong to the same plane, form the following three dimensional parallelepiped as shown. The volume of this parallelepiped can be found by taking the cross product of U and V, and taking the dot product of the result with the vector W. The bars here that you see denote absolute value since this expression contains the dot product of two vectors. This result is a scalar, which is referred to as the triple scalar product.